Blow-up problems for quasilinear reaction diffusion equations with weighted nonlocal source

In this paper, we investigate the following quasilinear reaction diffusion equations  (b(u))t = ∇ · ( ρ ( |∇u|2 ) ∇u ) + c(x) f (u) in Ω× (0, t∗), ∂u ∂ν = 0 on ∂Ω× (0, t∗), u(x, 0) = u0(x) ≥ 0 in Ω. Here Ω is a bounded domain in Rn (n ≥ 2) with smooth boundary ∂Ω. Weighted nonlocal source satisfies c(x) f (u(x, t)) ≤ a1 + a2 (u(x, t)) (∫ Ω (u(x, t)) dx )m , where a2, p, α are some positive constants and a1, m are some nonnegative constants. We make use of a differential inequality technique and Sobolev inequality to obtain a lower bound for the blow-up time of the solution. In addition, an upper bound for the blow-up time is also derived.


Blow-up problems for quasilinear reaction diffusion equations with weighted nonlocal source
Juntang Ding

Introduction
The blow-up problems to reaction diffusion equations has been extensively investigated by many researchers.Much of the work prior to the turn of the century is referenced in [1,9,10].
More recent work, we refer readers to [13][14][15][16][17][18]21].In practical situations, one would like to know whether the solutions blows up and if so, at which time blow-up occurs.Hence, finding bounds for blow-up time has become the focus of the researchers, especially the search for lower bounds of blow-up time.Since Payne and Schaefer [20] introduced a first-order inequality technique and obtained a lower bound for blow-up time, many authors are devoted to the lower bounds of blow-up time for various reaction diffusion problems, (see, for instance, Corresponding author.Email: djuntang@sxu.edu.cn[3][4][5][6][7]).We note that above mentioned studies mainly aimed at seeking lower bounds for blowup time of local reaction-diffusion equations.In this paper, we concern the reaction diffusion equations with weighted nonlocal source ∂u ∂ν = 0 on ∂Ω × (0, t * ), u(x, 0) = u 0 (x) ≥ 0 in Ω. (1.1) In (1.1), Ω is a bounded domain of R n (n ≥ 2) with smooth boundary ∂Ω, ν represents the unit normal vector to ∂Ω, u 0 (x) ∈ C 1 (Ω) is a nonnegative function satisfying the compatibility condition, t * is the blow-up time if blow-up occurs, or else t * = ∞.Weighted nonlocal source satisfies where a 2 , p, α are some positive constants and a 1 , m are some nonnegative constants.Set R + = (0, ∞).Throughout this paper, we assume that b is a function, and f is a nonnegative C(R + ) function.By maximum principles [22], we know that the classical solution u of (1.1) is nonnegative in Ω × [0, t * ).
For the information about the nonlocal reaction diffusion equations, we refer readers to [2,11,12,19,23].Fang and Ma [11] dealt with the following problems where Ω ⊂ R n (n ≥ 2) is a bounded star-shaped domain with smooth boundary ∂Ω, nonlocal source satisfies and a 2 , p, α, and m are positive constants.They derived conditions which imply the solution blows up in finite time or exists globally.Furthermore, upper and lower bounds for blow-up time are obtained.
As far as we know, there is little information on the bounds for blow-up time of problem (1.1).Motivated by the above work [11], we study the problem (1.1).Our results of this paper are based on some Sobolev type inequalities and differential inequality technique.In Section 2, when Ω ⊂ R n (n ≥ 2), we obtain a criterion for blow-up of the solution of (1.1) and get an upper bound for blow-up time.In Section 3, when Ω ⊂ R n (n ≥ 3), we derive a lower bound for blow-up time.An example is presented in Section 4 to illustrate our abstract results derived in this paper.

Blow-up solution
In this section, we establish conditions on data to ensure that the solution blows up at t * and obtain an upper bound for t * .To accomplish these tasks, we introduce the following auxiliary functions where u is the classical solution of (1.1).Our main result of this section is the following Theorem 2.1 Theorem 2.1.Let u be a classical solution of (1.1).We suppose that functions b, c, ρ, and f satisfy b (s) < 0, sρ(s) ≤ (1 + β)P(s), where β is a nonnegative constant.In addition, initial data are assumed to satisfy Then u must blow up at t * ≤ T in measure D(t) with Proof.It follows from Green's formula and (2.3) that Differentiating E(t), we get which with (2.4) imply E(t) > 0 and D (t) > 0 for all t ∈ (0, t * ).By the Hölder inequality, (2.5) and b (s) > 0 for s > 0, we obtain Using (2.3) and integrating by part, we lead to We combine (2.7) and (2.8) to derive that is Integrating (2.9) over [0, t], we have By (2.5), we can deduce This inequality can not hold for all t > 0. Hence, u(x, t) must blow up at some finite time t * in the measure D(t).Furthermore, we conclude from (2.11) .

Lower bound for blow-up time
In this section, we restrict Ω ⊂ R n (n ≥ 3).Our goal is to determine a lower bound for blow-up time t * .Here we impose the following constraints on data where a 2 , b 2 , p, q, γ are positive constants, a 1 , b 1 , m are nonnegative constants, p > 2q + 1, α = 2r(q + 1) − 2q, and parameter r is restricted by the condition We introduce two auxiliary functions In this section, we also need to apply the following Sobolev inequality (see [8, Theorem 2, p. 265]) for n ≥ 3, where C = C(n, Ω) is an embedding constant.The main result of this section is stated as follows.

Application
In this section, an example is presented to illustrate the applications of Theorems 2.1 and 3.1.

Example 4 . 1 .
Let u be a classical solution of the following problem:
and Xuhui Shen Abstract.In this paper, we investigate the following quasilinear reaction diffusion equations